Slightly surprising, but Chief Justice doesn't really have more power than any other Justice. Just a little more visible, particularly when swearing in the start of a new presidential term.
Not quite true.
The CHIEF Justice chairs the SCOTUS conference. There is power in this. When the Chief Justice chairs the conference, it is his/her prerogative to conduct the vote-taking. If the Chief Justice is in the majority, he/she assigns the opinion. If the Chief Justice is in the minority, the Justice with the most seniority in the majority opinion assigns the vote.
In Bob Woodward's
The Brethren, Woodward talked about a practice of Chief Justice Warren Burger that annoyed the liberals on the Court. When the conference convened and a final vote was to be taken, Burger would often "reserve" his vote for strategic purposes. (Customarily, the Chief Justice voted first.) The purpose of this was (A) to allow Burger to never be alone in an opinion (liberal critics said of Burger he was "Always to the Right, but never alone!"), and (B) able to control the assignment of all 5-4 and 6-3 opinions, with Burger often becoming a sixth vote in order to control the assignment of writing the opinion. This would allow Burger to assign the opinion to himself, or assign it to a less liberal Justice in order to blunt the effect of the case. (Burger reportedly try to assign opinions to Justices when he was in the minority, but the Conference rebelled against this.)
Imagine, if you will, John Roberts voting in the majority of
Obergefell, then assigning the opinion to himself. That's a "for example".
Roe v. Wade gives an example of what that would be like. The conference made it clear that they wanted to legalize abortion (at least the liberal majority did). Burger voted with the majority; he did so in order to control the assignment. He assigned the case to Harry Blackmun, who was then still considered a conservative, and his record to that point justified that conclusion. While I am not a fan of
Obergefell, its legal reasoning, and the dissents to the same, are fairly straightforward.
Roe was never that straightforward; there's a reason it has not survived.
There is a power benefit to being the First Among Equals. Burger was an effective Chief Justice even when he was a minority Chief Justice.
